WEBVTT
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As a warm-up towards finding
the distribution of the
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function of random variables,
let us start by considering
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the discrete case.
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So let X be a discrete random
variable and let Y be defined
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as a given function of X. We
know the PMF of X and wish to
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find the PMF of Y. Here's
a simple example.
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The random variable X takes the
values 2, 3, 4, and 5 with
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the probabilities given in the
figure, and Y is the function
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indicated here.
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Then, for example, the
probability that Y takes a
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value of 4.
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This is also the value of the
PMF of Y evaluated at 4.
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This is simply the sum of the
probabilities of the possible
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values of X that give
rise to a value of Y
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that is equal to 4.
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Therefore, this expression is
equal to the probability that
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X equals to 4 plus the
probability that
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X is equal to 5.
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Or, in PMF notation, we can
write it in this manner.
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And in this numerical example,
it would be 0.3 plus 0.4.
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More generally, for any given
value of little y, the
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probability that the random
variable capital Y takes this
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particular value is the sum of
the probabilities of the
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little x that result in
that particular value.
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So the probability that the
random variable capital Y,
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which is the same as g of X,
takes on a specific value is
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the sum of the probabilities
of all possible values of
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little x where we only consider
those values of
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little x that give rise to the
specific value, little y, that
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you're interested.
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Let us now look into the special
case where we have a
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linear function of a discrete
random variable.
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Suppose that X is described by
the PMF shown in this diagram,
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and let us consider the random
variable Z, which is defined
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as 2 times X. We would like
to plot the PMF of Z.
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First, let us note the values
that Z can take.
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When X is equal to minus
1, Z is going to be
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equal to minus 2.
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When X is equal to 1, Z is
going to be equal to 2.
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And when X is equal to 2, Z
is going to be equal to 4.
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This event that X is equal
to minus 1 happens with
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probability 2/6, and when that
event happens, Z will take a
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value of minus 2.
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So this event happens with
probability 2/6.
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With probability 1/6, X takes a
value of 1 so that Z takes a
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value of 2.
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And this happens with
probability 1/6.
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6 And finally, this last
event here happens
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with probability 3/6.
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We have thus found the PMF of Z.
Notice that it has the same
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shape as the PMF of X, except
that it is stretched or scaled
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horizontally by a factor of 2.
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Let us now consider the random
variable Y, defined as 2X plus
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3, or what is the same
as Z plus 3.
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With probability 2/6, Z
is equal to minus 2.
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And in that case, Y is going
to be equal to plus 1.
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And this event happens
with probability 2/6.
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With probability 1/6, Z takes
a value of 2 so that Y it
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takes a value of 5.
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And finally, with probability
3/6, Z takes a value of 4 so
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that Y it takes a value of 7.
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What we see here is that the PMF
of Y has exactly the same
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shape as the PMF of Z, except
that it is shifted to the
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right by 3.
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To summarize, in order to find
the PMF of a linear function
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such as 2X plus 3, what we do
is that we first stretch the
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PMF of X by a factor of
2 and then shift it
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horizontally by 3.
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We can also describe the PMF
of Y through a formula.
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For any given value of little
y, the PMF is going to be
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equal to the probability that
our random variable Y takes on
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the specific value little y.
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Then we recall that Y has been
defined in our example to be
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equal to 2X plus 3, so we're
looking at the probability of
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this event.
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But this is the same as the
event that X takes a value
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equal to y minus
3 divided by 2.
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And in PMF notation, we can
write it in this form.
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So what this is saying is that
the probability that Y takes
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on a specific value is the same
as the probability that X
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takes on some other
specific value.
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And that value here is that
value of X that would give
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rise to this particular
value little y.
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Now, we can generalize the
calculation that we just did.
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And more generally, if we have
a linear function of a
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discrete random variable X, the
PMF of the random variable
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Y is given by this formula in
terms of the PMF of the random
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variable X. The derivation
is the same.
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We use b instead the specific
number 3, and we have a
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general constant a instead
of the 2 that
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we had in this example.
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And this formula describes
exactly what we did
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graphically in our
previous example.
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This factor of a here serves to
stretch the PMF by a factor
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of a, and this term b here
serves to shift the PMF by b.